If \( \tan\alpha=\frac{1}{7} \) and \( \tan\beta=\frac{1}{3} \), Find \( \cos2\alpha \)
Question
If
\[ \tan\alpha=\frac{1}{7},\qquad \tan\beta=\frac{1}{3}, \]
then \(\cos2\alpha\) is equal to
(a) \(\sin2\beta\)
(b) \(\sin4\beta\)
(c) \(\sin3\beta\)
(d) \(\cos2\beta\)
Solution
Using the identity
\[ \cos2\alpha = \frac{1-\tan^2\alpha} {1+\tan^2\alpha} \]
Substituting \(\tan\alpha=\frac{1}{7}\),
\[ \cos2\alpha = \frac{1-\frac1{49}} {1+\frac1{49}} = \frac{48/49}{50/49} = \frac{24}{25} \]
Now find \(\sin2\beta\):
\[ \sin2\beta = \frac{2\tan\beta} {1+\tan^2\beta} \]
\[ = \frac{2\left(\frac13\right)} {1+\frac19} = \frac{2/3}{10/9} = \frac{3}{5} \]
Next,
\[ \cos2\beta = \frac{1-\frac19} {1+\frac19} = \frac{8/9}{10/9} = \frac45 \]
Using
\[ \sin4\beta = 2\sin2\beta\cos2\beta \]
\[ = 2\cdot\frac35\cdot\frac45 = \frac{24}{25} \]
Therefore,
\[ \cos2\alpha=\sin4\beta \]
Final Answer
\[ \boxed{\cos2\alpha=\sin4\beta} \]
Hence, the correct option is (b) \(\sin4\beta\).